On the best Hardy constant for quasi-arithmetic means and homogeneous deviation means

The aim of this paper is to characterize the so-called Hardy means, i.e., those means $M\text{\hspace{0.17em}⁣}:{\bigcup }_{n=1}^{\mathrm{\infty }}{\mathbb{R}}_{+}^n\to {\mathbb{R}}_{+}$ that satisfy the inequality $$\sum _{n=1}^{\mathrm{\infty }}M(x_1,\dots ,x_n)\le C\sum _{n=1}^{\mathrm{\infty }}{x}_n$$ for all positive sequences $(x_n)$ with some finite positive constant $C$. The smallest constant $C$ satisfying this property is called the Hardy constant of the mean $M$. In this paper we determine the Hardy constant in the cases when the mean $M$ is either a concave quasi-arithmetic or a concave and homogeneous deviation mean.